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SECONDARY 2 MATHEMATICS CLASSROOM · CHAPTER 1 · PROPORTION · MAP SCALES · DIRECT & INVERSE PROPORTION · G2/G3
Proportion and Map Scales: Find What Stays Constant Before You Calculate
In this classroom, you will not decide between multiplication and division by guessing from the wording. You will identify the relationship that remains true.
Secondary 2 proportion extends the multiplicative reasoning built in Secondary 1. A direct proportion preserves a quotient. An inverse proportion preserves a product. A map scale preserves a geometric ratio between representation and reality. The arithmetic is often short. The hard part is deciding which relationship the situation actually supports.
Classroom rule: name the quantities → identify what is fixed → choose the proportion model → calculate → preserve units → test the direction → verify the invariant → return the answer to context.
The current MOE G2 and G3 Mathematics syllabuses place map scales and direct and inverse proportion in Secondary Two. This classroom therefore treats them as the main Chapter 1 content while deliberately retrieving the ratio, rate and percentage foundations they depend on.
Official reference: MOE G2 and G3 Mathematics Syllabuses.
Navigate: Secondary 1 retrieval · direct proportion · inverse proportion · direct versus inverse · tables and graphs · map scales · area scale · modelling · misconception clinic · guided practice · assessment transfer · exit ticket.
Featured Answer: What Is Proportion?
Proportion describes a multiplicative relationship between quantities. In a direct proportion, one quantity is a constant multiple of another. In an inverse proportion, the product of the two quantities stays constant. A scale is another proportional relationship: it compares a represented length with a corresponding real length.
Direct proportion: quotient stays constant. Inverse proportion: product stays constant. Scale: corresponding lengths keep the same ratio.
How to Use This Classroom
- Attempt every Your Turn question before opening the answer.
- Write the invariant first: quotient, product or scale ratio.
- Keep labels and units visible.
- Use a direction check before exact arithmetic.
- Use tables or equations when the relationship becomes harder to hold mentally.
- Verify a direct proportion by checking the quotient.
- Verify an inverse proportion by checking the product.
- Verify a map calculation by reversing the scale.
- For area scales, square the length scale instead of applying it only once.
- Return later without the heading “direct” or “inverse” and decide the model yourself.
1. Retrieval: Ratio Is an Ordered Multiplicative Comparison
If A:B=2:3, then A/B=2/3. The order matters. B:A is 3:2, not 2:3.
2. Equivalent Ratios Preserve a Multiplicative Relationship
2:3, 4:6 and 10:15 are equivalent because both terms are multiplied by the same factor.
Adding the same amount to both terms does not generally preserve the ratio.
3. Part-to-Part and Part-to-Whole Are Different References
If there are 12 blue and 18 red counters:
- blue:red=12:18=2:3;
- blue:total=12:30=2:5;
- blue fraction of total=2/5;
- blue percentage=40%.
4. Teacher Model 1: Share a Total in a Ratio
Share 420 in the ratio 3:4.
Total parts=7. One part=420÷7=60.
Shares=180 and 240.
Check: 180+240=420 and 180:240=3:4.
5. Difference Problems Use the Difference in Ratio Parts
A:B=3:5 and B exceeds A by 24.
Difference=2 parts, so one part=12.
A=36, B=60.
6. Link Ratios by Matching the Shared Quantity
A:B=2:3 and B:C=4:5.
Make B the same: 8:12 and 12:15.
A:B:C=8:12:15.
7. Proportion Extends Ratio Into a Relationship Between Changing Quantities
Ratio often compares quantities at one moment. Proportion asks how quantities vary while some multiplicative relationship stays fixed.
Your Turn 1
- Share 560 in the ratio 3:5.
- A:B=4:7 and B−A=36. Find A and B.
- A:B=3:4 and B:C=2:5. Find A:B:C.
Answers
210 and 350. A=48 and B=84. A:B:C=3:4:10.
8. Direct Proportion: The Quotient Stays Constant
If y is directly proportional to x, then:
y=kx and therefore y/x=k for non-zero x.
9. The Constant k Is the Multiplier Connecting the Quantities
If y=7x, then every y-value is seven times its corresponding x-value. Here k=7.
10. Doubling x Doubles y in a Direct Proportion
If y=kx and x is multiplied by 2, then y is also multiplied by 2. Tripling x triples y. Halving x halves y.
11. Teacher Model 2: Find the Constant First
y is directly proportional to x. When x=6, y=42.
42=6k, so k=7. Therefore y=7x. When x=11, y=77.
12. Direct Scaling Gives the Same Answer
x changes from 6 to 11 by the factor 11/6. So y changes from 42 to 42×11/6=77.
13. Direct Proportion Is Stronger Than “Both Quantities Increase”
The relationship C=4+3n increases as n increases, but it is not a direct proportion because of the fixed 4. C/n is not constant.
14. A Direct-Proportion Graph Passes Through the Origin
For y=kx, x=0 gives y=0. The graph is a straight line through the origin.
15. A Straight Line Is Not Automatically Direct Proportion
y=3x+5 is linear, but because it does not pass through the origin it is not direct proportion.
16. The Units of k Matter
If y is cost in dollars and x is mass in kilograms, k may be dollars per kilogram. The constant is not merely a number; it describes the conversion between quantities.
17. Teacher Model 3: Unit Price as Direct Proportion
At a constant unit price of $4.50 per kilogram, C=4.5m. For 6 kg, C=$27. This model assumes no fixed fee, minimum charge or quantity discount.
18. Teacher Model 4: Production at a Constant Rate
A machine produces 45 units per minute at a constant rate. N=45t. At 8 minutes, N=360 units.
19. Direct Proportion Can Be Written as a Ratio Equality
If y/x is constant, then y₁/x₁=y₂/x₂. This gives a proportion equation connecting two corresponding pairs.
20. Cross-Multiplication Is a Consequence of Equality of Ratios
If a/b=c/d with non-zero denominators, then ad=bc. Use this as an algebraic consequence, not as a detached trick.
21. Teacher Model 5: Direct Proportion by Equal Ratios
8 units correspond to 28 outputs. How many outputs correspond to 15 units under direct proportion?
28/8=y/15. Therefore 8y=420, so y=52.5.
Your Turn 2
- y∝x and y=24 when x=8. Find k and y when x=13.
- A constant unit price gives $18 for 4 kg. Find the cost of 7.5 kg.
- Explain why y=5x+2 is not direct proportion.
Answers
k=3 and y=39. Unit price=$4.50/kg, so cost=$33.75. It has non-zero intercept 2, so y/x is not constant and the graph does not pass through the origin.
22. Inverse Proportion: The Product Stays Constant
If y is inversely proportional to x, then y=k/x and therefore xy=k.
23. Doubling x Halves y in an Inverse Proportion
If xy=k and x doubles, y must halve to keep the product fixed.
24. Tripling x Divides y by Three
Inverse proportion reverses multiplicative scale factors. That is different from simply subtracting the same amount.
25. Teacher Model 6: Find the Constant Product
y is inversely proportional to x. When x=4, y=15. Then k=xy=60, so y=60/x. When x=10, y=6.
26. Inverse Proportion Needs a Condition That Makes the Product Meaningful
For identical workers doing a fixed amount of fully shareable work at a constant individual rate, workers×days=constant worker-days.
27. Teacher Model 7: Fixed Work
Six identical workers complete a task in 15 days. Total work=90 worker-days. With 10 workers, days=90/10=9.
28. Direction Check: More Workers Should Mean Less Time
In the fixed-work model, an answer larger than 15 days for more workers should immediately be questioned.
29. Opposite Direction Does Not Automatically Prove Inverse Proportion
y=20−x decreases as x increases, but xy is not constant. This is a fixed-sum relationship, not inverse proportion.
30. Teacher Model 8: Fixed Distance
For a fixed distance d, speed×time=d. If speed doubles and distance is unchanged, time halves.
31. A 20% Increase in Speed Does Not Cause a 20% Decrease in Time
For fixed distance, multiplying speed by 1.2 multiplies time by 1/1.2=5/6. That is a decrease of 16⅔%, not 20%.
32. Teacher Model 9: Inverse Proportion From a Pair
y∝1/x and y=18 when x=5. k=90, so y=90/x. When x=12, y=7.5.
33. Combined Direct Changes Can Be Multiplicative
Three identical printers produce 240 pages in two hours at a constant rate. Five printers for four hours use the factor (5/3)(4/2)=10/3. Pages=240×10/3=800.
34. State the Assumption Behind a Worker or Machine Model
Inverse and direct models often assume identical productivity, unchanged conditions, enough resources and no fixed setup delay.
Your Turn 3
- y∝1/x and y=21 when x=6. Find k and y when x=14.
- Eight identical workers take 15 days. Find the time for 12 workers under a fixed-work model.
- A fixed journey takes 4 hours at 60 km/h. How long at 80 km/h?
Answers
k=126 and y=9. 10 days. Distance=240 km, so time=3 hours.
35. Direct and Inverse Proportion: Compare the Invariant
| Relationship | Equation | Invariant | If x doubles |
|---|---|---|---|
| direct | y=kx | y/x=k | y doubles |
| inverse | y=k/x | xy=k | y halves |
36. Build a Direction Check Before Calculating
- more items at constant unit price → more total cost;
- more time at constant production rate → more output;
- more workers for fixed work → less time;
- greater speed for fixed distance → less time.
37. Direction Is Necessary but Not Sufficient
Two quantities increasing together could follow y=x², y=3x+5 or many other rules. You still need evidence that the quotient is constant.
38. A Table Can Reveal the Invariant
For pairs (2,6), (4,12), (7,21), y/x=3 each time. This supports direct proportion. For pairs (2,30), (3,20), (5,12), xy=60 each time. This supports inverse proportion.
39. One Pair Alone Cannot Distinguish Every Possible Rule
The point (2,6) lies on y=3x, but it also lies on many other functions. The problem statement or several data pairs must establish the relationship.
40. Direct Proportion and Graphs
y=kx gives a straight line through the origin. The gradient is k when x and y are plotted on ordinary Cartesian axes.
41. Teacher Model 10: Read k From a Direct-Proportion Graph
A line through the origin passes through (4,18). k=18/4=4.5, so y=4.5x.
42. A Non-Zero Intercept Breaks Direct Proportion
A line such as y=4x+6 may represent a fixed charge plus a unit rate. Its gradient still has meaning, but the whole relationship is not direct proportion.
43. Inverse Proportion Does Not Produce a Straight Line on Ordinary Axes
y=k/x produces a curved graph for positive x. As x increases, y decreases while the product stays fixed.
44. Tables Are Often the Cleaner Secondary 2 Representation
When the main task is to distinguish direct and inverse proportion, checking quotient or product in a table can be clearer than relying on visual shape alone.
45. Teacher Model 11: Classify From a Table
| x | y | y/x | xy |
|---|---|---|---|
| 2 | 24 | 12 | 48 |
| 3 | 16 | 16/3 | 48 |
| 6 | 8 | 4/3 | 48 |
The product is constant at 48. The relationship is inverse proportion.
46. Map Scale Is a Proportional Relationship Between Representation and Reality
A scale of 1:25,000 means one unit on the map corresponds to 25,000 of the same unit in reality.
47. Scale Ratios Require the Same Unit on Both Sides
1 cm:25,000 cm is valid. 1 cm:250 m describes the same relationship, but it is not written as a pure numerical ratio until both are converted to a common unit.
48. Teacher Model 12: Interpret 1:25,000
1 cm on the map represents 25,000 cm in reality. 25,000 cm=250 m=0.25 km. So 1 cm represents 0.25 km.
49. Teacher Model 13: Map Distance to Real Distance
At scale 1:25,000, two points are 6 cm apart. Real distance=6×0.25=1.5 km.
50. Reverse Scale: Real Distance to Map Distance
At the same scale, 2 km appears as 2÷0.25=8 cm.
51. Keep Straight-Line Map Distance Separate From Route Distance
A measured straight segment on a map represents straight-line distance under the scale. A winding road or path can be longer. The question must state which route or measurement is intended.
52. Teacher Model 14: Scale 1:50,000
1 cm represents 50,000 cm=0.5 km. A 7 cm map distance represents 3.5 km.
53. Build the Conversion Chain Explicitly When Units Are Mixed
map cm → real cm → real m → real km.
Writing the chain prevents a scale factor and a metric conversion from being silently merged incorrectly.
54. Scale Can Also Be Written as “1 cm Represents …”
1:20,000 is equivalent to 1 cm representing 200 m. Move between forms according to what makes the calculation clearest.
55. Teacher Model 15: Find the Scale
4 cm on a drawing represents 1 km in reality. 1 km=100,000 cm. Scale=4:100,000=1:25,000.
Your Turn 4
- At 1:40,000, what real distance does 5.5 cm represent in km?
- At the same scale, what map length represents 3.2 km?
- 3 cm represents 750 m. Find the numerical scale.
Answers
1 cm=0.4 km, so 2.2 km. 3.2÷0.4=8 cm. 750 m=75,000 cm, so 3:75,000=1:25,000.
56. Area Scale Is the Square of the Length Scale
If every length is multiplied by k, every area is multiplied by k². This follows because area combines two perpendicular lengths.
57. Teacher Model 16: One Square Centimetre at 1:25,000
1 cm represents 0.25 km. Therefore 1 cm² represents 0.25×0.25=0.0625 km².
58. Teacher Model 17: Map Area to Real Area
At 1:25,000, a region has map area 4 cm². Each cm² represents 0.0625 km². Real area=4×0.0625=0.25 km².
59. Applying the Length Factor Once to Area Is a Dimensional Error
Area has two dimensions. A one-dimensional scale factor cannot be applied only once and still represent area correctly.
60. Scale 1:n Gives Area Scale 1:n²
At 1:100, the area scale is 1:10,000. At 1:500, the area scale is 1:250,000.
61. Teacher Model 18: Reverse Area Scale
At 1:20,000, 1 cm represents 0.2 km. Therefore 1 cm² represents 0.04 km². If a real region is 0.6 km², its map area is 0.6÷0.04=15 cm².
62. Area Scale Connects Directly to Similarity
Later similarity work generalises the same idea: length scale k produces area scale k² and, for solids, volume scale k³.
63. Do Not Confuse Map Area With Route Length
Area and distance answer different geometric questions even when they come from the same map.
Your Turn 5
- At scale 1:50,000, what real area does 3 cm² represent in km²?
- At scale 1:20,000, a real region is 0.32 km². Find its map area in cm².
- Explain why a 1:100 length scale does not produce a 1:100 area scale.
Answers
1 cm=0.5 km, so 1 cm²=0.25 km² and 3 cm²=0.75 km². At 1:20,000, 1 cm²=0.04 km², so map area=8 cm². Area contains two scaled lengths, so the factor is squared: 1:10,000.
64. Proportion Is a Model, Not a Default Assumption
Real situations often contain fixed charges, capacity limits, changing rates, setup times or coordination delays. Use direct or inverse proportion only when the conditions support it.
65. Teacher Model 19: Fixed Fee Breaks Direct Proportion
An invented service charges $6 fixed plus $4 per unit. C=6+4n. The unit component is proportional to n; total cost is not.
66. Capacity Limits Can Break a Machine Model
Doubling machines may fail to double output if the power supply, raw material or shared conveyor becomes the bottleneck.
67. Coordination Delays Can Break a Worker Inverse Model
Twice as many people do not always halve real completion time. Inverse proportion applies when the problem’s conditions make total work the dominant fixed quantity.
68. Teacher Model 20: Select the Model, Then Solve
An invented printing job uses identical printers. Three printers produce 540 pages in 6 minutes at unchanged individual rates. Three printers for six minutes=18 printer-minutes, so the rate is 30 pages per printer-minute. Five printers for ten minutes=50 printer-minutes. Output=1500 pages.
69. Sensitivity Check: Change One Quantity and Predict Direction
Before exact arithmetic, ask whether the output should rise, fall or stay unchanged. This direction check catches many direct-versus-inverse swaps.
70. Boundary Check: What Happens at Zero?
A direct model y=kx gives y=0 at x=0. If a proposed “direct proportion” still has a non-zero total at x=0, inspect whether there is a fixed term.
71. Misconception Clinic: “Both Increase, So It Is Direct Proportion”
Repair: check whether y/x is constant and whether the graph passes through the origin.
72. Misconception Clinic: “One Goes Up and One Goes Down, So It Is Inverse”
Repair: check whether xy is constant.
73. Misconception Clinic: Use the Same Percentage Change in Reverse
A 20% increase followed by a 20% decrease does not return to the starting value because the base changes.
74. Misconception Clinic: Multiply Time by the Worker Factor
In fixed-work inverse proportion, more workers require less time. Reverse the factor.
75. Misconception Clinic: Treat a Fixed Charge as Part of Direct Proportion
A fixed intercept means total output or cost does not scale from zero.
76. Misconception Clinic: Apply a Length Scale Once to Area
Area has two dimensions, so the length factor must be squared.
77. Misconception Clinic: Mix Units Inside a Scale Ratio
Convert corresponding lengths to the same unit before forming or simplifying the numerical scale.
78. Misconception Clinic: Map Distance Equals Travel Route Distance
A straight map measurement is not automatically a winding route length.
79. Misconception Clinic: Cross-Multiply Before Identifying Corresponding Quantities
A proportion equation is only useful when the ratios compare corresponding quantities in the same order.
80. Misconception Clinic: One Data Pair Proves a Proportion
One pair can fit infinitely many rules. Use the stated condition or additional pairs.
81. Guided Practice A: Direct Proportion
- y∝x, y=35 when x=7. Find y when x=12.
- y∝x, y=18 when x=4.5. Find k.
- A constant unit rate gives 96 items in 8 minutes. Find output in 13 minutes.
Solutions
k=5, so y=60. k=4. Rate=12 items/min, so 156 items.
82. Guided Practice B: Is It Direct?
- y=8x.
- y=8x+5.
- Pairs (2,10),(4,20),(7,35).
- Pairs (2,10),(4,18),(7,30).
Answers
Direct. Not direct. Direct with k=5. Not direct because y/x is not constant.
83. Guided Practice C: Inverse Proportion
- y∝1/x, y=24 when x=3. Find y when x=9.
- Ten workers take 18 days. Find the time for 15 workers under fixed work.
- A fixed journey takes 5 h at 72 km/h. Find the time at 90 km/h.
Solutions
k=72, so y=8. 180 worker-days÷15=12 days. Distance=360 km, time=4 h.
84. Guided Practice D: Classify Direct or Inverse
- Number of identical tickets and total price with no fixed fee.
- Number of workers and time for fixed shareable work.
- Speed and time for a fixed distance.
- Time and distance at fixed speed.
Answers
Direct. Inverse. Inverse. Direct.
85. Guided Practice E: Map Distance
- At 1:30,000, find real distance for 8 cm.
- At 1:30,000, find map distance for 4.5 km.
- 5 cm represents 2 km. Find scale.
Solutions
1 cm=0.3 km, so 2.4 km. 4.5÷0.3=15 cm. 2 km=200,000 cm, so 5:200,000=1:40,000.
86. Guided Practice F: Map Area
- At 1:40,000, find real area represented by 2.5 cm² in km².
- At 1:10,000, find map area for 0.18 km².
Solutions
1 cm=0.4 km, so 1 cm²=0.16 km² and 2.5 cm²=0.4 km². At 1:10,000, 1 cm=0.1 km and 1 cm²=0.01 km², so map area=18 cm².
87. Guided Practice G: Mixed Model Selection
A production process uses four identical machines for 6 hours to produce 1440 units. At unchanged individual rates, find output from six machines for 10 hours.
Worked solution
Original machine-hours=24, so 60 units per machine-hour. New machine-hours=60, so output=3600 units.
88. Guided Practice H: Mixed Percentage and Inverse Proportion
A fixed trip takes 6 hours at speed v. Speed rises by 25%. Find the new time.
Worked solution
New speed factor=1.25, so time factor=1/1.25=0.8. New time=4.8 hours, a 20% decrease.
89. Challenge Practice: Direct, Linear or Inverse?
Classify each relationship and justify using an invariant or intercept:
- y=6x.
- y=6x+4.
- xy=48.
- y=30−x.
Answer
Direct proportion. Linear but not direct proportion. Inverse proportion. Fixed-sum linear relationship, not inverse proportion.
90. Challenge Practice: Find a Scale From Area Information
A 9 cm² map region represents 0.36 km². Assuming the map uses one uniform linear scale, find how many kilometres 1 cm represents.
Worked solution
1 cm² represents 0.04 km². Therefore 1 cm represents √0.04=0.2 km. That is 200 m=20,000 cm, so the scale is 1:20,000.
91. Challenge Practice: Two-State Ratio Into Equation
Two amounts are in ratio 3:5. After 10 is added to each, the ratio becomes 2:3. Find the original amounts.
Worked solution
Let amounts be 3k and 5k. (3k+10)/(5k+10)=2/3. Then 9k+30=10k+20, so k=10. Original amounts are 30 and 50.
92. Assessment Method: State What Is Fixed Before Calculating
- direct proportion → constant quotient;
- inverse proportion → constant product;
- map scale → constant corresponding-length ratio;
- area scale → square of length scale.
93. Assessment Method: Check Direction First
If x grows, predict whether y should grow, shrink or depend on more information. This catches many model-selection errors before arithmetic begins.
94. Assessment Method: Keep Corresponding Quantities in the Same Order
When using ratio equations, do not write one side as output/input and the other as input/output.
95. Assessment Method: Carry Units Through Scale Work
Write cm→m→km explicitly when the scale and answer use different units.
96. Assessment Method: Square Before Applying an Area Scale
If the linear factor is k, area changes by k².
97. Assessment Method: Test the Invariant
- direct → recompute y/x;
- inverse → recompute xy;
- scale → convert the result back to the representation.
98. Assessment Method: Do Not Force a Proportion When Information Is Insufficient
If the problem gives no constant-rate or proportionality condition, a unique proportional answer may not be justified.
99. Oral Classroom Check
- What stays constant in direct proportion?
- What stays constant in inverse proportion?
- Why does direct proportion pass through the origin?
- Why is y=3x+5 not direct proportion?
- Why is “one up, one down” not enough to prove inverse proportion?
- How do you verify an inverse-proportion answer?
- What does scale 1:25,000 mean?
- Why must scale ratio units match?
- Why is area scale squared?
- What condition can break a worker inverse-proportion model?
100. Exit Ticket
- y∝x and y=32 when x=8. Find y when x=15.
- y∝1/x and y=20 when x=6. Find y when x=15.
- Ten workers take 12 days. Find time for 15 workers under fixed-work assumptions.
- At scale 1:50,000, find real distance represented by 9 cm.
- At the same scale, find map distance for 6 km.
- At the same scale, find real area represented by 2 cm².
- Explain why y=7x+3 is not direct proportion.
- A fixed-distance journey becomes 30% faster. State whether the travel time decreases by 30%, and explain.
Exit-ticket solutions
k=4, so y=60. k=120, so y=8. Total work=120 worker-days, so 8 days. 1 cm=0.5 km, so 4.5 km. 6÷0.5=12 cm. 1 cm²=0.25 km², so 2 cm²=0.5 km². Non-zero intercept 3 means the quotient y/x is not constant. No: time is multiplied by 1/1.3≈0.7692, a decrease of about 23.1%.
101. Homework: Retrieval, Variation and Transfer
Layer 1 — Retrieval
- state y=kx and y/x=k;
- state y=k/x and xy=k;
- explain one difference between a direct proportion and a straight line with a fixed intercept;
- state the meaning of 1:n scale;
- state why area scale is 1:n².
Layer 2 — Variation
- three direct-proportion questions;
- three inverse-proportion questions;
- two classification questions;
- three distance-scale questions;
- three area-scale questions;
- one model-limit explanation.
Layer 3 — Transfer
Create one real-world situation that is genuinely direct proportion, one that is genuinely inverse proportion, and one that looks proportional at first but contains a fixed term. State the invariant or reason in each case.
102. The Seven-Day Return Cycle
- Day 0: complete teacher models and guided practice.
- Day 1: solve one direct, one inverse and one map-scale question.
- Day 3: classify four unlabeled relationships and solve one area-scale problem without notes.
- Day 7: complete the exit ticket with changed data and explain every model choice aloud.
103. A 60-Minute Teaching Lesson
- 5 minutes: ratio retrieval.
- 15 minutes: direct proportion and constant k.
- 15 minutes: inverse proportion and constant product.
- 10 minutes: direct-versus-inverse classification.
- 10 minutes: map distance and area scale.
- 5 minutes: exit ticket.
104. A 90-Minute Teaching Lesson
- 10 minutes: Secondary 1 ratio/proportion retrieval.
- 20 minutes: direct proportion, tables and graphs.
- 20 minutes: inverse proportion and model conditions.
- 15 minutes: mixed classification.
- 15 minutes: map scales and area scales.
- 5 minutes: modelling limits.
- 5 minutes: exit ticket and return date.
105. The Full Direct-Proportion Routine
name quantities → confirm direct condition → calculate k=y/x → write y=kx → solve → verify quotient → interpret.
106. The Full Inverse-Proportion Routine
name quantities → confirm fixed-product condition → calculate k=xy → write y=k/x → solve → verify product → interpret.
107. The Full Map-Scale Routine
identify map/real quantity → align units → apply length scale in the correct direction → convert units → reverse-check.
108. The Full Area-Scale Routine
find linear factor → square it → apply to area → preserve square units → reverse-check by square root if needed.
109. Connect Back to Secondary 1
This chapter grows directly from Secondary 1 Chapter 3: Ratio and Proportion, Chapter 4: Percentages, Chapter 5: Rate, Speed and Unit Conversion, and Chapter 8: Coordinates, Linear Functions, Graphs and Gradient.
110. Specialist Companion
For narrower specialist repair and additional ratio/rate/percentage examples, use Secondary 2 Mathematics Learning Guide | Ratio, Proportion, Rate and Percentage.
111. Why This Chapter Matters for the Rest of Secondary 2
Proportion reappears inside similarity, maps, rates, graphs, algebraic models and percentage reasoning. The habit of asking what stays constant will become increasingly useful as expressions and equations become more complex.
112. Ready for Chapter 2?
You are ready to continue when you can do all of the following without prompts:
- distinguish ratio from proportion;
- find and use a direct-proportion constant;
- find and use an inverse-proportion constant;
- distinguish direct proportion from a linear relation with a fixed intercept;
- distinguish inverse proportion from any relationship that merely decreases;
- use tables to test quotient and product invariants;
- interpret direct proportion graphically;
- convert map and real distances in both directions;
- form a scale from corresponding lengths;
- square the length factor for area scale;
- state assumptions behind worker, machine and journey models;
- verify answers using the invariant and direction.
If one item is weak, return to the smallest section that owns it and solve a changed example. If all are stable, continue to Chapter 2, where proportional reasoning gives way to deeper algebraic structure: expansion, formulae, identities and controlled symbolic transformation.
